At a workshop in Alameda County last month, I made my standard request for classroom teachers to help me make good on my New Year’s resolution. I assumed all the teachers there taught middle- or high-school so I said yes to every teacher who invited me. Later, I’d find out that one of them taught fourth grade.

As a former high school math teacher, this was NIGHTMARE MATERIAL, Y’ALL.

I mean, what do fourth graders even look like? I’m tall, but do I need to worry about stepping on them? What do they know how to do? Do they speak in complete sentences at that age? Clearly, what I don’t know about little kids could fill libraries.

Children are teenagers are adults. Don’t let me make too much of my one hour of primary education experience, but I was struck hard by the similarities between all the different ages I’ve taught. People of all ages like puzzles. They respond well to the techniques of storytelling. Unless they’re wildly misplaced, they come to your class with some informal understanding of your lesson. They appreciate it when you try to surface that understanding, revoice it, challenge it, and help them formalize it. I handled a nine year-old’s ideas about a jar of Skittles in exactly the same way as I handled a forty-nine year-old’s ideas about teaching middle schoolers.

I asked them if I had left money on the table, if I had missed any opportunities to challenge and chase student thinking. They brought up an interesting debate from the end of class, a real Piagetian question about whether a different jar would change the number of Skittles. (It wouldn’t. The number of packages was fixed.) I had asked students to imagine another jar, but my hosts thought the debate demanded some manipulatives so we could test our conjectures. Nice!

Also, Spencer told me that when he asks students to talk with each other, he asks them to share out their partners’ thinking and not their own. That gives them an incentive to tune into what their partners are saying, rather than just waiting for their own turn to talk. Nice! As a secondary teacher, I felt like a champ if I asked students to talk at all. Spencer and his primary colleagues are onto some next-level conversational techniques.

Primary students have more stamina than I anticipated. No doubt much of this credit is due to the norms Mr. Spencer has set up around his “Problem Solving Fridays.” But I’ve frequently heard rules of thumb like “children can concentrate on one task for two to five minutes per year old.” These kids worked on one problem for the better part of an hour.

The pedagogy interests me more than the math.

I think elementary math pedagogy is more interesting than secondary but I don't know if I can get excited about the math.

This sentiment still holds for me after today. I just find algebra more interesting than two-digit multiplication. I’ll try to keep an open mind. Today was not an interesting day of math for me, though it was a very interesting day of learning how novices learn and talk about math.

I’m probably not wacky enough for this work. Mr. Spencer greeted his students by calling out “wopbabalubop!” to which they responded “balap bam boom!” Really fun, and I don’t think you can teach that kind of vibe.

Loads of algorithms, and none of them “standard.” Graham’s 3-Act modeling task asks students to multiply two-digit numbers. I saw an area model. I saw partial products. Students used those approaches flexibly and efficiently. They were able to locate each number in the world when asked. I didn’t see anyone carry a one. Everyone should settle down. This is great.

I expected the experience would either kill me or convince me I should have taught primary students. This one fell somewhere in the middle. I’m excited to return someday, and I was happy to witness the portability of big ideas about students, learning, and mathematics from adult education to high school to elementary school.

I remember my first venture in elementary school after teaching ninth grade algebra and eighth grade math for four years. I was curious about younger students and my friend invited me into her third grade class. I can’t remember anything about the lesson I taught, but what I do remember is that I made a student cry. He had done something that I thought was inappropriate behavior and I must have responded pretty harshly. Hey, I was used to teaching older tough kids and I had never thought about modulating my response. It wasn’t my finest hour and I was devastated. My friend helped me through the experience and I even went back. After then I learned other ways to talk with younger students and became more and more fascinated about how they formed their conceptions . . . and misconceptions . . . about mathematical ideas. I’m hooked.

34 Comments

I remember my first venture in elementary school after teaching ninth grade algebra and eighth grade math for four years. I was curious about younger students and my friend invited me into her third grade class. I can’t remember anything about the lesson I taught, but what I do remember is that I made a student cry. He had done something that I thought was inappropriate behavior and I must have responded pretty harshly. Hey, I was used to teaching older tough kids and I had never thought about modulating my response. It wasn’t my finest hour and I was devastated. My friend helped me through the experience and I even went back. After then I learned other ways to talk with younger students and became more and more fascinated about how they formed their conceptions . . . and misconceptions . . . about mathematical ideas. I’m hooked.

Hey, want to come to the fifth grade class I teach in San Francisco?

Joe vignolini

So my life changed watching jk-4 math classes.. I saw great things.. now as a jk-12 math chair I am working with all teachers seeing classes, and more. What is more fun than teaching bc calculus then going to teach a grade 3-4 stem activity with zip lines… a math challenge.
Those who have not tried need to go, observe, get involved.
We use INVESTIGATIONS… which is amazing to see kids learn….
My single advice to hs teachers… they are kids who like scratch and sniff stickers…
come to northern Virginia dan near D.C. If you want

joe vignolini

agree that the research is mixed. Probably due to a few lurking variables. 1. the education of teachers generally began with traditional algorithms, 2. the students may not experience a “throughline” of preschool to high school of the non traditional algorithms, 3. parents often teach what they know – so they introduce the algorithms while alternative ones are being done. Fear of falling behind. 4. Students see a “rule” as easier-faster to learn as they compartmentalize learning into “facts” and many more.
This is an interesting paper – one of many that discuss the shift….

but I think we need to realize that “traditional algorithms” are NOT traditional – rather a product of mass education in a “classroom” of rows and desks. See the Chinese tools for multiplication, the Ethiopian coffee bean methods, the lattice etc… It seems that what we used to do when we taught individuals is “contemporary” and what we do when we teach “all in a group” is “traditional. Why don’t we go back to what helped make the pyramids, navigate the world etc without the tricks… focus on the History of numbers and learn ways that not only worked for individuals, but also “created” the math we see today…

Congrats on a successful experience outside of your comfort zone! This year, I have been working in elementary schools primarily and initially, I was hesitant like yourself. Not only was I nervous to work with younger students, but I was also unsure that I had anything meaningful to offer at that level.

Five months later and I’m loving the younger grades. I’ve learned a ton from my elementary colleagues and I’m starting to make connections I never realized between primary and secondary math.

interesting!

My current focus has been how to create tasks that start with really basic concepts and extend them up to algebra. Looking at math through this lens has really opened my eyes to how we might better ensure students can connect the dots all the way through.

The task starts as a simple subitizing problem, then an array, then order of operations and moves on to algebra. Excited to try and think up some more that can start with such a low floor and high ceiling.

Congratulations on surviving elementary school! Let me know when you’re in Southern California and want to make your debut in first grade. I think you would be astounded with their discourse! I like to ask them “Find out what your partner thinks about _____.” Then, like you said, ask them to share what their partner said when thinking about other students’ work. The more I learn about the progressions and study Van de Walle’s Teaching Students Mathematics, the more excited I get about the math we learn in class because I understand more about where it is going. Maria Blanton has shared a ton about developing algebra and algebraic thinking all throughout elementary school. Super interesting work.
I was allowed to guest teach in fifth grade last week. That was pretty exciting, and it again opened my eyes to how important content knowledge is because I wasn’t prepared for how one student responded and I think I left money on the table there that could have benefited the whole class. As always, thanks for sharing! Please, come visit any time!

Wait…. Could “a different jar [of the same size]… change the number of Skittles?” You’re saying it wouldn’t. Why not? Can I please re-invigorate this debate? Isn’t is reasonable to assume that even if the same number of packets is consistent, that there is *some* small amount of variability with the number of Skittles between the packages.

Dan Meyer

We had a fixed number of packages with a variable number of Skittles between the packages. But the total Skittles is conserved. Swap one container out with another and the number of Skittles across those packages isn’t going to change.

Ok… lemme think about this. A fixed number of packets with a variable number of Skittles between the packages.

Idea 1: Total in jar will be fixed, since what? Since number of packets is large enough that the average in packet is realized?
Idea 2: Total in jar will have some possible variability, since what? Since each individual packet might have some variability?

This is a great mathematical/statistical question. I’m still thinking about this…

As a third grade teacher of math, I’m thrilled to read that your observations match my own experiences. I’m inspired to invite our high school math teachers into the classroom for a bit of dabbling in next-level conversations and pedagogy. More of this, please!

interesting!

I enjoy seeing the scope and sequence of how our elementary math teaching builds into middle and high school problem-solving. It’s a question I ask myself frequently. “How will secondary math classes use the problem-solving techniques I’m instilling in our students?”

Angela Cooper

I actually went the other way: I taught third grade for several years before I switched to high school and now I teach Math 1 but that experience was invaluable. I think we, adults, underestimate what children are capable of. I spend a lot of time in my high school math classes developing math vocabulary, something that, for whatever reason, I thought my third graders couldn’t handle. I know now how wrong I was about that.

interesting!

We miss out on cultivating real interest in math when we underestimate what young children can, and more importantly want to, do. I know now that it was easier to encourage curiosity in my third graders than in my high school students. I didn’t capitalize on that at the time and I regret it now.

Like Marilyn, I started as an 8th grade math teacher and quickly realized I wanted to move down in age. It seemed like understanding of multiplication/division and fractions (rational numbers in general I guess) was the basis of EVERYTHING I taught in pre-algebra/algebra, so I landed in the year that students first dig deep into those – 3rd grade. It’s terrifying and awesome.

Nice vignette about your 4th grade teaching experience. Former High School math teacher here, and in my role as PD provider, I’ve had multiple opportunities to model lessons in K-12 classrooms. I still remember listening to Kinders argue which is taller, a house or a tree. It was great! Turning to talk to a partner, these little people justified their decision using evidence about their own home, house, apartment, and the trees surrounding it. It was great! Sadly, during a wonderful chat about percents of numbers with some 7th grade students, their classroom teacher said, “I had no idea they could think about percent and understand it in that way.” UGH. The students had just completed ‘finding a percent of a number’ worksheet in which they basically used a sequence of key strokes on their non-scientific calculator. Double UGH! I love talking to youngsters about THEIR ideas….age 6, 12, or 18!

“I think elementary math pedagogy is more interesting than secondary but I don’t know if I can get excited about the math.”

My first reaction to that comment was whole hearted agreement. I never really gave elementary math much thought until I had kids. That paired with the amazing folks in Mtbos has given me a new appreciation and fascination at what elementary teachers do. I currently teach 7th grade math because, for me, it’s always been that perfect pairing of math that I can find enjoyable and students that I can find enjoyable…but in researching elementary math for my kids, I wonder what it would be like to teach younger students, although I’m pretty sure I have neither the energy nor creativity to survive (I can’t make a bulletin board to save my life).

My second reaction was to wonder why elementary pedagogy is more developed. I had a few thoughts:
Elementary math is more broad: It’s all about number sense and foundations of mathematical understanding. That is incredibly important and is often where people lay blame when students grow up with poor number sense, so it’s where we (mathematicians) tend to focus our research.

Elementary teachers talk to each other more than secondary teachers…maybe. This is absolutely based on personal experience and I’m sure it’s not the case everywhere, but I tend to see elementary teachers ask for help more often and more open to discuss their failures. Secondary teachers tend to be the experts so there isn’t as much need to discuss different approaches. Chances are, this isn’t overly true of your readers and I’m only observing secondary and elementary teachers from here in the middle…but since it seems more teachers discuss math pedagogy in elem, it’s more developed and interesting.

On a personal level, I’ve thoroughly enjoyed researching elementary pedagogy because there’s something special about seeing the young mind at work. The brain of a 6 or 9 year old (the ages of my kids) is an amazing thing and I could sit and listen to them talk about math forever.

joe vignolini

Dan
“I think elementary math pedagogy is more interesting than secondary but I don’t know if I can get excited about the math.”
It takes a little time – but honestly – even after teaching Calc III, AP courses, the grade 3-4 content can be exciting. especially if students are allowed to engage and not following “standard algorithms” and you see where it leads. Much like teaching Alg I knowing the Calculus ahead is exciting.
THE ONLY thing is that – I do not have the patience for being a “homeroom teacher” and doing it all day with the same kids every day… Elementary teachers have amazing skills there…

So my life changed watching jk-4 math classes.. I saw great things.. now as a jk-12 math chair I am working with all teachers seeing classes, and more. What is more fun than teaching bc calculus then going to teach a grade 3-4 stem activity with zip lines… a math challenge.
Those who have not tried need to go, observe, get involved.
We use INVESTIGATIONS… which is amazing to see kids learn….
My single advice to hs teachers… they are kids who like scratch and sniff stickers…
come to northern Virginia dan near D.C. If you want

I spent my first teaching prac with a Year 1/2 class and my colleague teacher was teaching the class about arrays. She turned everything into a mathematical opportunity and extended every discussion to include addition AND multiplication. Someone’s birthday? An array of muffins. Class prize? A chocolate block. Just finished art? The materials would reappear on the floor in yet another array! Each was a different size which made me think about factors and the work that she put into extending the arrays towards multiples of 12 (which have more factors and therefore more arrangements). I was fascinated and very thankful that such young minds were exposed to such broad discussions! Again, no algorithms were necessary.

Julian Gilbey

Hi, great story! I had a similar experience: I taught high school math in the UK (6th-12th graders) for 10 years, and last year guest taught every year group in a local elementary school (K-5th grade), doing investigational problems. It was terrifying but highly enlightening. I did an activity about sorting and classifying shapes based on their properties with the kindergarten class (UK Year 1, 5-6 year olds), similar to an NRICH activity (http://nrich.maths.org/5997), and was amazed that they stayed focused and engaged on the activity for a whole hour. We can so easily underestimate what younger students are capable of!

Boy, I remember my first time teaching elementary school too. I was teaching mostly high school at a K-12 school, but landed a 4th grade classroom (just the math) in my second year. As good as I felt about my work with the ninth, tenth and eleventh graders, working with the fourth graders was a huge challenge. It ended up being the major focus of my work that year, especially since I was designing most of the curriculum myself, and all my instincts from the high school level didn’t seem to work at that age.

Fortunately, I was bailed out by another teacher, who had designed a bunch of curriculum and offered it to me to use. (This was 2003.) A big part of what I learned that first year was about having two or three shorter tasks to do, rather than one large one, which was my typical strategy for high school.

It was a challenging year. I made a lot of mistakes, and grew a lot as a teacher. What’s interesting is that now, fourth grade is one of my favorite grades to teach, and—as you say—I absolutely expect fourth graders (and younger kids too) to be able to focus for 60 minutes or more if the subject matter is compelling.

Thanks for the post, and for sharing this @gletchy 3-act! And for underlining the purposeful wackiness of elementary teachers. It’s like they have superpowers.

Not much to add to what’s been said above, only that it must be gratifying to see the 3-act construct you created for high schoolers play out in elementary school. Big thanks to Graham for creating so many of those tasks. And if you really want an experience, try doing a 3-act with kindergarteners!

Everyone has their right to an aesthetic preference for particular areas/topics/levels of math. The cool thing about math is that (almost) every topic can be really fun to investigate because it is open to a deeper exploration of pattern, structure, and connections to other areas.

A weakness of math education is that again, almost every topic can be presented in a way that is closed, shallow, isolated, and boring.

Also, I’m going to outbid @Joe Schwartz: many of Graham’s 3 acts (and many of the Desmos activities Dan has highlighted here) work well with 3 or 4 year olds! We don’t always get to the intended ending point and we don’t always follow the “standard” path through the activity, but we always have a fun and rewarding mathematical discussion.

Joe

Dan Meyer

There seems a divide, sometimes unhealthy, btwn the world’s of ET and HS (understanding, bias, instruction, content, etc). Thoughts on bridging that divide?

Great question. There are certain tasks, like the Pool Border problem, that have hooks down in primary instruction and can extend up to secondary. I wonder if lesson studies ever cross that many grades.

Apologies, I should have better clarified the word ‘divide’ better. I haven’t checked back past few days. Flu got the better of me.

When I say divide, I think I mostly mean in terms of instruction. I’m in elementary and am about to make a wide sweeping generalization (so forgiveness if it’s too big). I think many of us, not all, in elementary, if we’re being honest, do not understand math to the level we should. We don’t always understand how early childhood to elementary to adolescence to high school builds mathematically.

I think our instruction really matters, but I couldn’t tell you specifically how it builds for later mathematical development. Likewise, I couldn’t do what you do in the classroom. I could run a lesson, but I couldn’t interact with the math. Just being honest. But I think our elementary instruction could become more robust if we could. I’m not sure if it works in the opposite direction? HS to elementary?

So my question was asking how we begin to bridge that divide? Hope that makes a bit more sense.

joe vignolini

Tyler,
This I understand. I am a JK-12 Math Chair and have meetings with teachers at all levels. I would say that even the “best trained” math teacher at any level, has trouble in connections. Knowing math is not the same as understanding and being able to teach it. We use Investigations, which is a very detailed course and flow. Teachers who spend the time reading it, following it and teaching – often find the inquiry is there, they can lead a good conversation and like math.

So not my to answer the question – I think the ONLY way to bridge a divide – is to have “flow”. This means a consistent “text” and series flow JK-12. The issue here is everyone wants to add their own “spices.” And some schools are way too big. We have 1000 students JK-12 and for the most part, there is a flow. The divide is because we operate in “silos” and as “islands” (by schools, by subjects, by grades)

I would say that what I generally see in Elementary school – using manipulatives, activities, etc… is essential.. it should “last all the way to grade 12 and beyond”. So it does work in both directions. I actually see “new” ideas about HS teaching and wonder – because I saw in in the earlier grades. If you can – observe HS and MS teachers – but NEVER i mean NEVER – think “I need to prepare them for that teacher or that challenge”…

only think – I need to prepare my kids to think and adapt.

Lisa Scranton

As someone who is currently pursuing my Single Subject Teaching Credential, I have been entirely focused on teaching high-school and middle-school students. Having recently transitioned from teaching in a high school to a middle school, I have been quite surprised (in a positive way) to observe the impacts and changes in teaching that have occurred within the last few years. At the high school, everything was much more straightforward, and it was sometimes difficult to discern the concepts that students struggled with, because they were expected to know so much. After all, these individuals were fundamentally still products of the pre-Common Core era, wherein procedural work and memorization were viewed as sufficient to demonstrate understanding. Once I began teaching at the middle school, however, I immediately noticed how the standards had affected students’ learning. I realized that my seventh-grade students were first-grade students when the standards were implemented, and as a result, had been exposed to a different way of thinking since- essentially- their educational/academic birth. These students seem more willing to explore mathematical reasoning, and ask questions on deeper conceptual levels than the high-school students did.

Nevertheless, I could have an exaggerated perception about these generalizations, but the distinctions are, at the very least, quantifiable.

I was enthralled to read in your post that these fourth-grade students seem to have a firm grasp on number sense, and that you didn’t notice anyone “carrying” digits during the multiplication process. I feel like if I had had this exposure and opportunity when I was in the fourth grade, I would feel more confident teaching at the elementary-school level. The new standards are a filter, in a way, to ensure that teachers actually *understand* mathematical concepts, instead of simply teaching procedures. It wasn’t until I started university that I began to fully comprehend the “why”s and “how”s of mathematical mechanics. In fact, I found myself guilty of using a procedure when investigating Graham Fletcher’s 3-Act Task. In the first video, I made an estimate of the number of Skittles in the jar. But then in the second video, I used the information provided about the number of Skittles in each bag, and the number of bags used, to adjust my estimate. How did I do this? I took the product of 14 (the number of Skittles in a bag) and 58 (the number of bags). Yes, I fell for this trick! It took a moment to realize that the number of bags was fixed, but the number of candies within each bag is likely to differ. It’s these conceptual habits that should be fostered within students, beginning at the youngest possible age. I believe that children are naturally curious about these activities as it is, and already have an inherent sense of wonderment. But somewhere along the way (before the new standards), this desire to learn was hampered by robot-like procedures. These children are the future, and I feel very inspired knowing that they seem to be on an optimal path!

Dan Meyer

Thanks for sharing, Lisa. I had to call out one of your paragraphs with a “whoa” above. I hadn’t given enough thought to the difference between students whose entire education was in the Common Core era versus those who have had to adapt. Very provocative. Looking forward to checking in on these ideas more at CSUEB later this semester.

I’ve had the opportunity to teach in a variety of elementary classrooms this year using 3-Acts, and I’m loving it! My own classroom experience has been at the secondary level, so this has been a good stretch for me. First graders are very different from my usual AP Calculus students in some ways, and not so different in others. :)

100% yes

The challenge I’ve had is anticipating how students will approach a task and misconceptions they’ll have (which are sometimes about life and not about math), and planning my responses to their struggle so that it stays productive.

So far this year I’ve taught in 1st, 2nd, 4th, 5th and 6th grade classrooms, sometimes using the same task with different grade levels which has been very interesting to see how students see and approach things differently. It’s been a great learning experience.

The challenge I’ve had is anticipating how students will approach a task and misconceptions they’ll have (which are sometimes about life and not about math), and planning my responses to their struggle so that it stays productive.

Right! Our most interesting struggle was around the different jar question. The math was more straightforward. That was interesting.